Weighted $L^2$ Holomorphic functions on ball-fiber bundles over compact Kähler manifolds

Abstract

Let $\tilde{M}$ be a complex manifold and $\Gamma$ be a torsion-free cocompact lattice of $\text{Aut}(\tilde{M})$. Let $\rho :\Gamma \to SU(N,1)$ be a representation and $M:=\tilde{M}/\Gamma$ be an $n$-dimensional compact complex manifold which admits a holomorphic embedding $ı$ into $\Sigma :={\mathbb{B}}^N/\rho (\Gamma )$. In this paper, we investigate a relation between weighted $L^2$ holomorphic functions on the fiber bundle $\Omega :=M{\times }_{\rho }{\mathbb{B}}^N$ and the holomorphic sections of the pull-back bundle ${ı}^{-1}(S^m{T}_{\Sigma }^{\ast })$ over $M$. In particular, $A_{\alpha }^2(\Omega )$ has infinite dimension for any $\alpha >-1$ and if $n<N$, then $A_{-1}^2(\Omega )$ also has the same property. As an application, if $\Gamma$ is a torsion-free cocompact lattice in $SU(n,1)$, $n\ge 2$, and $\rho :\Gamma \to SU(N,1)$ is a maximal representation, then for any $\alpha >-1$, $A_{\alpha }^2({\mathbb{B}}^n{\times }_{\rho }{\mathbb{B}}^N)$ has infinite dimension. If $n<N$, then $A_{-1}^2({\mathbb{B}}^n{\times }_{\rho }{\mathbb{B}}^N)$ also has the same property.